Is that a picture of their enclosure on the right? If so, it is is not safe for them. Chicken wire does not keep predators out, they can rip or chew through it. Also, if you are in an area with RHDV2 (see the pinned post about this), housing outside is dangerous if they aren’t vaccinated. There are some examples of safe and appropriate outdoor housing ideas here: https://binkybunny.com/bunnyinfo/safe-outdoor-runs/
Most people who house outdoors have a combo of a secure hutch and a run. A rabbit isn’t likely to really get the right type of exercise on a harness, and they can injure themselves very easily if they panic while wearing one.
Do you know if your partner is mostly allergic to the hay, or to them? Many people with allergies have found that orchard grass is a lot less allergenic than timothy. You can also make simple hepa filters using a MERV 13 air filter and a box fan. I have several of these set up and they seem to really help (my sister in law is very allergic and spent some time in our house without issues).
If all of that doesn’t help, then I agree that rehoming may be the best option.
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The answers provided in this discussion are for general guideline purposes only. The information is not intended to diagnose or treat your pet. Seek the advice of your veterinarian or a qualified behaviorist.